Numerical computations of the electric-arc heating and anode melting were carried out within the framework of the two-dimensional unsteady mathematical model. The influence of the viscous interaction “plasma−melt”, surface tension forces, electromagnetic forces, and gravitational convection on the formation of the hydrodynamics of the anode melt was considered.
Key words: electric arc, numerical modeling, anode unsteady heating and melting.
Introduction
In the first part of the paper, a two-dimensional mathematical model and the nu-merical technique for computing the characteristics of the unsteady electric arc, includ-ing the conjugate heat exchange of the electric arc plasma flow with the treated product (anode), were considered. The arc characteristics were computed since the ignition mo-ment until the passage to the stationary regime. In the present paper, the results of the numerical computation of the unsteady process of the anode heating and melting are presented.
1. Steel anode with a low sulfur content
First of all, we elucidate the methodic aspect of the numerical solution of the hy-drodynamic part of the problem, which is due to the discrete character of the computa-tional region. When solving a difference problem it is customary to assume that the melt-ing starts if the anode temperature Т reaches or exceeds the melting temperature Tmelt near the surface of the anode near-axial region on two grid layers in axial direction and on four layers in radial direction. As a matter of fact, even when the condition Т ≥ Tmelt is satisfied on the anode surface and, consequently, the melt forms, it appears impossible to carry out the computation of the melt hydrodynamics only on a single grid layer. Note that the above region is a rectangle of the size (in the z, r plane) 0.1 × 0.3 mm at the grid
step value, for example, ∆ = 0.1 mm. Besides, although the equation for the energy bal-ance with respect to the enthalpy is used for computation of the anode thermal state, the temperature, which is a more clear characteristic in the given case, will be used at the discussion of the results.
Consider an anode of steel of the grade SUS 304 with a low (40 ppm) sulfur con-tent. In this case, the temperature gradient of the surface tension of the melt has the nega-
tive value ∂Г/∂Т = −4.48·10−4 N/(m·K). The thermophysical properties and the transfer coefficients of the anode material are assumed in accordance with the data of the work [1].
Since on the melt surface, as a rule, ∂T/∂r < 0, the Marangoni force acting on a unit
surface of the melt FMr = Г
T
∂∂
T
r
∂∂
has the positive sign and, consequently, in the given
case it contributes to the melt flow in the positive direction along r, that is from the near-axial region to the weldpool periphery.
By the moment of time t = 10 ms, the arc characteristics have already passed to the stationary regime (see [2]), however, the maximum temperature of the anode surface amounted to 804 K, and only by the time t = 80 ms it reached the melting temperature value, remaining invariable until t = 150 ms due to the consideration of the latent heat of melting. As a result, within the framework of the above-accepted assumption, at the given external parameters of the discharge, the process of the anode “numerical” melting starts in 200 ms after the arc initiation. This figure is obviously slightly overes-timated in comparison with the actual time of the melting start.
Near the anode melt surface, the arc plasma flow diverges in radial direction and be-cause of the viscous interaction makes the surface layers of the melt move in the same direction.
The Marangoni force, which is insignificant on the most part of the melt surface (≈ 15 N/m2), also contributes to the flow of the melt upper layers from the center to the periphery of the pool, at 7 < r < 7.5 mm reaches its maximum values (≈ 270 N/m2) (see Fig. 1). Such a distribution of FMr is due to the temperature pro-file on the melt surface: the temperature gradient has the highest value right on the pool periphery. Note that the Marangoni force takes a negative value because of a positive temperature gradient in radial direction in the interval 0 < r < 2 mm, although it is very small. A rapid increase in FMr on the pool periphery has nearly no effect on the velocity of the melt surface layer. That is just the viscous interaction with the arc plasma whose velocity reaches its maximum value v ≈ 23 m/s at the distance of r ≈ 1 mm is in the given case the main reason for the melt motion. The melt velocity on the surface does not ex-ceed ≈ 55 cm/s.
The maximum temperature of the melt surface realizes at some distance from the axis (at r ≈ 2 mm) rather than in the near-axial region, where the arc plasma tempera-ture is the highest, and amounts to Т ≈ 2 kK. Furthermore, the radial profile of tempera-ture is nonmonotonous ⎯ in the interval 0 < r < 2 mm the melt surface temperature slightly increases and only then decreases (see Fig. 1). Such a distribution of temperature is due to the convective removal of the heat from the near-axis region in radial direction.
As can be seen in Fig. 2, the energy of the neutralization of electrons Qφ = jφa,
the conductive heat transfer Qλ = −λ∂Т/∂z from the arc plasma to the anode, and the en-ergy of electrons making the arc current Qe = 5/2 kTe j/qe are the main mechanisms of
the anode heating. The anode cooling by radiation εakST4 from the surface is by about
an order of magnitude lower, and the Joulean heat release j2/σ is even more less ⎯ by two orders.
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Figure 3 shows the evolution of the melt hydrodynamics at different moments of
time in the interval t = 1−20 s (the scale of vectors in the picture of vector fields is not
supported). It is seen that two toroidal vortices arise in the weldpool ⎯ the large one and, on the pool periphery, the small vortex with the same right-screw (clockwise) rota-tion direction. The highest velocity of the melt flow realizes on the surface and near the solid side wall of the weldpool (see Fig. 4).
The melt flow character contributes to the convective heat transfer in radial direction from the near-axis region of the pool to periphery. As a result of such a pattern of the melt flow, the largest depth of melting 1.8 mm is reached on the pool periphery the radius of which amounted to 7.5 mm by the moment of time t = 20 s (Fig. 5) rather than in the near-axis region.
In about after five seconds of the arc burning, the variation of the dynamic and thermal characteristics of the melt are insignificant (see Fig. 6). The fluctuations of the axial velocity component u of the melt call attention, whereas they are practically absent for the radial velocity component v. These fluctuations are apparently due to a discrete variation of the weldpool sizes.
The grid Peclet number (Pe = ρVср∆/λ is the ratio of the intensities of the con-
vective and conductive heat transfer) on the melt surface and in the central region of the toroidal vortex exceeds unity by several times, which points to the predominance of the convective heat transfer on the conductive heat transfer.
The axial temperature distribution on the anode axis (see Fig. 6) has the hori-zontal interval at 1 < z < 3.2 mm, which is due to the latent melting heat. The further computed temperature distribution has
Fig. 1. Radial distributions of temperature Т, radial velocity component vliq and the Marangoni force FMr on the melt surface, varc is the plasma velocity at a distance of the grid step from the melt surface. Steel with a low sulfur content.
Fig. 2. Radial distributions of specific thermal fluxes Qϕ, Qλ, and Qe on the anode surface. Steel with a low sulfur content.
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a linear character because the coefficient of thermal conductivity of the anode material is assumed constant.
A somewhat unusual geometric shape of the weldpool calls attention: it has the maximum depth on the periphery rather than on the axis.
2. Steel anode with high sulfur content
For the steel melt with a high (220 ppm) sulfur content we have ∂Г/∂Т =
= 2.08·10−4 N/(m·K), and since ∂Т/∂r < 0 on the most part of the melt surface, then
the Marangoni force FMr = Г
T
∂∂
T
r
∂∂
will have the negative value. Thus, FMr contributes
in the given case to the melt flow in the negative radial direction ⎯ from the periphery to the near-axis region of the weldpool, and, consequently, impedes the flow caused by
the viscous interaction “plasma−melt”.
Fig. 3. Vector field of the velocity U of the anode melt at different moments of time t. Steel with a low sulfur content.
Fig. 4. Contours of the melt velocity V = 2 2u v+ at t = 20 s. Steel with a low sulfur content.
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An analysis of computational results shows that in the time interval t = 1−20 s,
the processes of heat and mass transfer are, on the whole, qualitatively and quantitatively
Fig. 6. Temporal dependence of the maximum values of the radial v, axial u velocity components and melt temperature T, axial distribution of temperature on the anode axis at t = 20 s.
Steel with low sulfur content.
Fig. 5. Temperature field Т of the anode at different moments of time t. Steel with a low sulfur content.
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identical with the foregoing case with the low sulfur content (right for this reason, Fig. 7
presents some numerical characteristics only for t = 20 s). The largest melting depth
and weldpool radius are equal to 2 mm and 7.1 mm, respectively (Fig. 7). An insignifi-
cant difference is observed only in the melt hydrodynamics on the weldpool periphery:
the small vortex has the left-screw (counter clockwise) rotation direction due to
the negative value of the Marangoni force.
The Marangoni force takes a negative very small value (≈ −5 N/m2) nearly on
the entire melt surface, however, on the pool periphery at r ≈ 6 mm it increases (in its
absolute value) up to ≈ 70 N/m2. Because of this the melt velocity on the pool periphery
also takes the negative value ≈ −15 cm/s, but it is positive on the remaining pool surface
and reaches ≈ 50 cm/s. Inside the weldpool, the maximum velocity of the melt flow
amounts to ≈ 28 cm/s.
Fig. 7. Vector field of the velocity U; temperature field Т of the anode melt by the moment t = 20 s; distribution of temperature Т, radial velocity component vliq and the Marangoni force FMr
on the melt surface.
Steel with a high sulfur content.
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3. Comparison of computed results with experimental data
It follows from the above-presented results that the weldpool characteristics are very close to one another for the anode with low and high sulfur content, whereas the data of the experiment and computations [1] show a significant difference.
Figure 8 shows the experimental and computed profiles of the anode weldpool [1]. It follows from experimental results that for a low and high sulfur content, the largest depth of the anode melting amounts to 1.82 mm and 4.3 mm, respectively, and the weld-pool radius amounts to 6.09 mm and 5.09 mm. It is worthwhile noting that besides a significant difference in the pool depth (by the factor of about 2.5), its geometric shape is qualitatively different.
For a steel with low sulfur content, the comparison of computed (see Figs. 3 and 5) and experimental data (see Fig. 8a) gives, on the whole, a fair agreement both in the geometric shape of the weldpool and in its sizes. However, the results of computing the weldpool shape and sizes differ significantly from experimental data of [1] for the steel with a high sulfur content (cf. Fig. 7 and 8b). The reasons for this may be differ-ent (we will assume that the mathematical model and the computational algorithm do not contain serious errors). A discrepancy in the results may be due, for example, to an in-sufficiently accurate technique for computing the viscosity coefficient on the control volume face passing between two neighboring grid lines one of which belongs to the melt surface, and the other belongs to the arc plasma. At the discussion of a similar technique, the authors of the work [3], referring to the limited size of the book, present only some general discussions. The author of the work [4] pronounces much more defi-nitely noting that there is in this question “… no complete clarity at present”. Despite a relatively long age of edition [4] one should recognize that the state-of-the-art has not improved significantly. In a number of works, the authors suggest considering the mechanism of the viscous friction at the molecular level, which, however, implies the difficulties of a different nature and, as a consequence, the objections of other
Fig. 8. Experimental profile of the weldpool [1], computed vector field of the velocity U and temperature field Т of the melt [1], t = 20 s.
a ⎯ low content of sulfur, b ⎯ high content of sulfur.
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researchers (see [5]). We recall that the technique of computing the coefficients on the control volume faces by the mean harmonic formula[6] is used in the present work. The given technique undoubtedly reflects qualitatively correctly the physics of the process, however, it possibly needs a refinement in the quantitative respect, which needs separate research.
An inaccurate information about thermophysical properties and transfer coeffi-cients of the arc plasma (mainly the viscosity coefficient [7]) and the anode material, including too a small value of ∂Г/∂Т of the temperature gradient of the surface tension coefficient of the melt may be another reason for the discrepancy of the results. It is, besides, possible that the computed value of the plasma radial velocity component near the melt surface is too high and, as a consequence, too a high momentum transfer from the plasma flow to the melt. As a result, the viscous interaction “plasma−melt” domi-nates excessively over the Marangoni force. In this connection, the additional model computations were carried out, first, with an increased value of ∂Г/∂Т and, second, at an invariable quantity ∂Г/∂Т = 2.08·10–4 N/(m·K) with an artificially underestimated value of the radial velocity component v of the plasma velocity in the immediate neighborhood of the melt surface.
An analysis of the results has shown that with an increase in ∂Г/∂Т from 2.08 up to 4.19·10–4 N/(m·K), there are no significant changes in the weldpool, the heat and mass exchange processes are identical with the ones considered above (see Fig. 7). However, at the value ∂Г/∂Т = 4.2·10–4 N/(m·K), one observes a qualitative restructuring of characteristics (we will call conventionally this value in the following text a “high” value). Already by the moment of time t = 0.5 s, two vortices of approximately equal intensity with opposite rotation directions form in the weldpool (Fig. 9). The right-screw vortex is due to viscous interaction “plasma−melt”, the left-screw vortex (further the Marangoni vortex for brevity) is due to the effect of the Marangoni force. Sufficiently quickly, already by the moment of time t = 0.55 s, the Marangoni vortex dominates in the entire pool region and contributes to convective heat transfer into the pool depth and, thereby, to an increase in the melting depth. The melt flow character and heat exchange do not change qualitatively subsequently ⎯ the Marangoni vortex dominates nearly in the entire pool volume (except for a narrow near-axial region on the melt surface), and by the moment t = 20 s, the pool depth and radius reach the values 5.5 mm and 5.1 mm, respectively (Fig. 10).
As compared to the foregoing variants, the velocity V of the melt flow decreases significantly, by 2−3 times, which causes a reduction of the convective heat transfer in-ward the pool. This leads to a significant increase in temperature Т of the melt near-axis
region ⎯ up to ≈ 2.5 kK and in tem-perature gradient (in its absolute value) in the radial direction. As a conse-quence of this, the Marangoni force increase by 2−3 times (in its absolute value) (Fig. 11).
A sharp boundary of the quantity
∂Г/∂Т ≈ 4.2·10–4 N/(m·K), at which there
Fig. 9. Vector field of the velocity U of the anode melt at different moments of time t (the computation with a “high” ∂Г/∂Т).
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occurs a passage from the flow formed mainly by the viscous interaction
“plasma−melt” (see the velocity vector field U in Fig. 7) to the flow formed by the Ma-rangoni force (see the velocity field U in Fig. 10). In the course of computations, we have not succeeded in obtaining some averaged stationary pattern of the melt flow. Such a melt flow appears to be unstable.
Comparing the computational results at a “high” value of ∂Г/∂Т we note (see Figs. 8b and 10) that the agreement in the computed and experimental data on the weldpool geometric shape and sizes is quite satisfactory.
Model computations with a reduced radial plasma velocity near the melt surface showed that the results at vmodel = 0.73·v and 0.72·v are the most typical.
The melt flow evolution at vmodel = 0.73·v in the interval of time t = 0−3 s is shown
in Figs. 12 and 13, which also show the model vmodel and computed values of the radial
Fig. 10. Velocity vector field U and temperature field Т of the anode melt by the moment of time t = 20 s (computation with a “high” ∂Г/∂Т).
Fig. 11. Distributions of temperature Т, radial velocity component vliq, and the Marangoni force FMr
on the melt surface (computation with a “high” ∂Г/∂Т).
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velocity component v of plasma near the melt surface. It is seen that by the moment of
time t ≈ 0.5 s, a toroidal vortex of the right-screw rotation direction forms in the weld-
pool, which is due to the viscous interaction “plasma−melt”. By the moment of time
t ≈1 s, the melt flow pattern changes significantly: as a result of the Marangoni force action, a toroidal vortex of the left-screw rotation direction forms on the pool periphery,
which takes by the moment t ≈ 2 s already the entire pool region. Further by the moment
Fig. 12. Velocity vector field at different moments of time t (model computation at νmodel = = 0.73 ν).
Fig. 13. Velocity vector field at t = 3 s, the profiles of the computed (ν) and model (νmodel = = 0.73 ν) gas velocity near the anode surface.
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of time t ≈ 2.2 s, the melt flow pattern again changes: as a result of the viscous interac-
tion “plasma−melt”, a toroidal vortex of the right-screw rotation direction again forms
and enhances, which dominates nearly in the entire region of the weldpool at t ≈ 3 s.
After that, no significant changes in flow character are already observed ⎯ the resul-tant stationary flow regime is similar to the one already considered above in the case of an anode with a high sulfur content (see Fig. 7).
A qualitatively different pattern of the melt flow is observed at a very insignificant
(about 1 %) reduction of the plasma radial velocity near the melt surface ⎯ at vmodel =
= 0.72·v. At the moments of time t = 0.5; 1; and 2 s, the melt hydrodynamics is similar to
the foregoing variant (see Fig. 12), however, starting from t ≈ 2 s, the Marangoni force
dominates over the viscous interaction “plasma−melt”, and this relation does not change subsequently. For this reason, Fig. 14 shows the computed data only for the final moment of time t = 20 s, when the processes have passed to the stationary regime. The compari-son of computed results shows that in this case, the agreement in numerical and experi-mental data on the weldpool geometric shape and sizes is quite satisfactory (see Fig. 8b and 14).
As in the case of the increase in ∂Г/∂Т, a sharp boundary of the quantity vmodel,
at which there occurs a passage from the flow formed by the viscous interaction
“plasma−melt” to the flow formed by the Marangoni force, calls attention. By varying vmodel we have not succeeded in obtaining the averaged steady flow pattern of the melt.
The conducted model computations are of methodical interest despite an arbitrary diminution of v and increase in ∂Г/∂Т and may be useful at the discussion of the peculi-arities of real physical processes. Furthermore, a reduced value of v and increased value of ∂Г/∂Т do not go beyond the framework of real values, which enables one to speak of at least qualitatively correct description of the physics of processes by the mathematical model.
The graphical presentation of the results of computing the melt hydrodynamics in the work [1] is unfortunately such (see Fig. 8) that it is rather difficult to represent
Fig. 14. Vector field of the velocity U and contours of temperature Т at the moment of time t = 20 s (model computation at νmodel = 0.72 ν).
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the general melt flow pattern and to perform the comparison. For a steel with a low sul-fur content, the agreement in the numerical results of [1] and of the present work, which concern the pool shape and sizes, is nevertheless satisfactory on the whole. But for a steel with a high sulfur content, the results of computing the anode temperature dif-fer significantly, and the hydrodynamic peculiarities of the melt are qualitatively different.
4. Influence of individual factors on the heat and mass exchange processes in the weldpool
It appears to be interesting to consider the influence of individual factors on the formation of the melt flow of the weldpool. We recall that the mathematical model
accounts for the viscous interaction “plasma−melt”, electromagnetic forces, gravitational convection, and the Marangoni convection.
To estimate the contribution of some factor of interest the computations were car-ried out at the “switched off” remaining factors in the computer code, which affect
the melt hydrodynamics. The computations were done in the time interval t = 1−20 s, the external parameters of the problem are invariable (see [2]).
An analysis of computational results showed that the processes of heat and mass exchange in the weldpool are qualitatively identical for each individual in-
vestigated factor in the time interval t = 1−20 s, therefore, the computed data are pre-sented in subsequent figures only for the time t = 20 s when the stationary regime is practically achieved. Besides, in some cases the distributions of characteristics for dif-ferent variants proved to be qualitatively close to the distributions, which were already considered above, therefore, these distributions are not presented in the figures to avoid the repetition.
The distributions of characteristics, which are due to the viscous interaction “plas-
ma−melt” only, practically coincide with similar distributions of the original variant,
in which all the factors are taken into account (see Figs. 3−6). The quantitative differ-ence in the results does not exceed several percents, which points to the predominant
role of the viscous interaction “plasma−melt” over the remaining three factors.
The influence of the Marangoni convection was considered by the example of
a low (∂Г/∂Т = −4.58·10−4 N/(m·K)) and high (∂Г/∂Т = 2.08·10−4 N/(m·K)) sulfur content
in the anode material. In the first case, the Marangoni force acts in the positive radial direction, and, as the analysis of the results showed, its influence is practically equivalent
to the viscous interaction “plasma−melt”, which was considered above in this Section, and even the quantitative differences are not so significant. The melt velocity values
are slightly less (by ≈ 20 %) in comparison with the viscous interaction “plasma−melt”, and, as a consequence, the convective heat removal is less, and the anode surface tem-
perature is higher (by ≈ 10 %). The weldpool shape and sizes coincide completely.
In the case of a high sulfur content in the anode material, the Marangoni force acts in the negative radial direction, and the heat and mass exchange processes in the weld-pool change significantly. They are close in qualitative respect to the ones considered
above (see Fig. 10 ⎯ the computation with a “high” ∂Г/∂Т and Fig. 14 ⎯ the model computation at vmodel = 0.72·v). The melt flow velocity on the pool surface does not ex-
ceed ≈ 10 cm/s, however, inside the pool it increases up to ≈ 20 cm/s, the maximum melt
temperature amounts to ≈ 2340 K.
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As a result of the effect of electromagnetic forces, a toroidal vortex of the left-screw rotation direction forms in the entire weldpool volume (Fig. 15). The flow charac-ter contributes to the heat transfer in axial direction from the upper hottest layers of the pool inward the pool. At the same time, the melt flow velocity is relatively low ≈ 4.5 cm/s, and the grid Peclet number Pe =ρVс
р ∆/λ does not exceed unity in the entire
weldpool volume. Because of a weak convective heat removal, the melt temperature on the pool surface does not drop below ≈ 2.9 kK.
The influence of the gravitational convection is very insignificant (Fig. 16), which is explained by a very weak dependence of the density of the melt on its temperature ⎯ the difference in the melt density values does not exceed one percent. Because of a low velocity of the melt flow (≈ 0.3 cm/s) the convective heat removal from the hot surface into the melt is practically absent (Ре << 1), which causes, in its turn, a high temperature of ≈ 3 kK of the surface. It is, however, not clear to what extent this temperature value corresponds to reality because a further growth of temperature is bounded in the com-puter code by the condition Т < 3 kK. It appears that under such conditions of the anode heating, it is necessary to account for the anode material evaporation. The model
Fig. 15. Contours of the electric current I, vector fields of electromagnetic forces Fmag and veloc-ity U in the melt (upper figures); contours of temperature Т and velocity V (lower figures). The effect of electromagnetic forces only.
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nevertheless reflects qualitatively correctly the role of the gravitational convection ⎯ a hotter melt in the upper near-axial region of the pool is displaced by a colder melt from the lower region of the pool.
Conclusions
Numerical computations of the electric-arc heating and melting of the anode were conducted within the framework of the two-dimensional unsteady mathematical model.
In the considered range of external parameters of the problem, the main mecha-nisms of the anode heating are the energy of electrons neutralization, conductive heat transfer from the arc plasma to the anode, and the energy of electrons composing the arc current. The cooling of the anode by radiation from the surface is insignificant, and the contribution of the Joulean heating is very small.
The main factors formulating the hydrodynamics of the anode melt are the viscous interaction “plasma−melt” and the negatively directed Marangoni force. The both fac-tors counteract one another and cause a qualitatively different character of the anode melt flow, geometric shape, and sizes of the weldpool. The influence of a positively directed Marangoni force, electromagnetic forces, and gravitational convection is very insignificant. There exists a sharp boundary, at which there occurs a passage from the flow formed mainly by the viscous interaction “plasma−melt” to the flow formed by the nega-tively directed Marangoni force. A certain stationary pattern of the melt flow will seem-ingly be unstable.
References
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Fig. 16. Vector field of the velocity U and temperature field Т in the melt. The effect of gravita-tional forces only.
